 An investor may be required to choose an investment among the available alternative investment options. But to choose a particular investment, it is desirable to evaluate the risk and return associated with each investment alternative. Now, how to calculate this return and risk associated with the particular investment option? An investor needs to learn this risk return trade-off competition. How can we define return on investment? A return on investment is a change in the investor's wealth that results from holding an investment over a particular period of time. That change in the wealth occurs due to two reasons. The first reason is the future cash flow in the form of dividend or interest. And the second change is the price appreciation that may be in terms of negative or positive. This means that the price of particular investment may be increased or decreased in the days to come. So, due to this change in the investment's principal price, the wealth of the investor may change. We have two types of returns. The first is the historical returns. These are the returns that an investor has earned in the past or the investment has experienced in the past over a particular period of time. The second type of return is the expected returns. These are the returns that can be computed as a weighted average over the longer period of past returns experienced by a particular investment. There emerged another concept that is holding period return. Holding period returns are the returns that are experienced by an investor. Holding a particular investment for a particular period of time in the past. Now, how to measure historical returns? In fact, historical returns are the returns that can help in evaluating the expected risk and return trade-off on a particular investment or multiple options available on alternative investments. To measure historical returns, we need to have ending value of investment and opening value of investment and that when we divide ending value of investment over its beginning value, the resulting figure is basically the holding period return. And if the holding period return is greater than 1, this means the returns are positive and that another means that the investment has created value for its owner. On the other side, if holding period returns are less than 1, this means that the return is negative and this another means that the investment has not created wealth for its owner. We can transform holding period returns into percentage and then we will be in fact computing holding period yield or HPY. Now, to compute holding period yield, we need to deduct 1 as a principle from the holding period return. If we have investment horizon over a multiple period of time, then we can determine analyzed holding period return. To analyze the holding period return, we just use the power of number of periods for which the investment is the hold. We have an example in this regard, which says that an investment costing $250 two years earlier is worth $350 now. Now to determine its holding period return, we just divide the ending value of $350 over the opening value of $250 and the resulting holding period return is 1.40. Now to analyze this 1.40, we have the power for two years and that is 1.1832. This is the analyzed holding period return. Now, if we want to determine the holding period yield of this value, then to determine this analyzed holding period yield, we will be deducting 1 from this holding period return. So, if we deduct 1 from 1.183, the resulting value is 18.32%. So, the analyzed holding period yield on this particular investment is 18.32% per annum. Similarly, we have another example which says that an investment costing $1000 two years earlier is worth $750 now. So, when we determine holding period return, it comes to 0.75 and if we analyze this holding period return, it comes to 0.866. Now, when we get the analyzed holding period yield, the resulting value is negative 13.40. So, this means in the first example, our holding period return is greater than 1. So, there is an increase in the wealth by 18.32%. And in the second example, our holding period return is less than 1. And the analyzed holding period yield now says that we have lost our investment by 13.40%. The other measure we have, the main historical returns are the realized return. Remember, an investment may get high return for a certain period of time and low return for a certain period of time. This means that there may be positive return for a particular time period and there may be negative return for a particular time period. And the analyst needs to consider both of these returns in his or her financial analysis. So, we can determine mean returns using these all types of returns. This means that mean return is basically the past return of past rate of return experienced by an investment. And it is expected that this return will be continued for an extended period of time in the future. Mean return or the average returns are the best way to measure returns on a particular investment. We can compute historical returns for a single investment and as a portfolio investment, both. To determine single investments mean historical return, we can compute them using arithmetic mean and geometric mean. When we use arithmetic mean, basically this is the sum of annual holding period yields, which is divided by the number of years. This means that we have total periods holding yields and we have a total period. And when we divide the total holding period yield over n, we get the arithmetic mean or the average holding period yield. To determine the geometric mean, basically it is the nth root of the product of the holding period returns for n years. And then we deduct this value one from this value. So, this means that we have the product of all holding period returns to the nth root minus one. For that purpose, we have an example here, we have a three years of time period. And for year one, we have beginning value of 100 and ending value of 115. The ending value of year one is the opening value of year two. Similarly, the ending value of year two is the opening value of year three. Using these opening and ending values, we have for each year the holding period return. Using this holding period return and the year, we get the holding period yield. Now we can use this holding period yield to determine arithmetic mean, which is 5% here. And when we use this holding period yield in our geometric means model, we get a geometric mean return of 3.353%. So, we see that our arithmetic mean is 5%, which is greater than the geometric mean of 3.35%. In fact, arithmetic mean is biased upward. But for that purpose, we assume that investment assets or investments longer period performance is perfect to measure. That is our prime purpose. Let's see an example, which says that a security that increases in price from $50 to $100 during year one and drops back to $50 during year two. Then the annual holding period yield, if we get for year one, it is one and for year two, it is minus 0.50. If we use these two values through in the arithmetic mean model, we get an arithmetic mean or mean average of 25%. This investment in fact brought no change in wealth and this means there is no return. As in the year one, we have a gain in the year two, we have a loss. So apparently there is no return, but the arithmetic mean still is given a 25% rate of return. But when we use these holding period yields in the geometric mean model, we see that the resulting figure is 0. This means that through the usage of geometric mean, we feel there is no return. This means that we are at break even. So the answer of 0% rate of return, this actually accurately measures the fact that there is no change in the owner's wealth through this investment after a period of two years. Geometric mean is considered as a superior tool over the arithmetic mean in determining long period mean returns. Because geometric mean indicates a compounded rate of return using the ending value of investment against its beginning value. If we compare geometric mean with the arithmetic mean, we see that if the rate of return remains the same for all of the time periods held for the investment, then the geometric mean is equal to the arithmetic mean. And if the rate of return is varying over the time horizon of the investment, then the geometric mean will be lesser than the arithmetic mean. As we have seen in our earlier example, where the geometric mean was 3.83 against the arithmetic mean of 5%. This shows that the larger annual changes in the rate of returns of investment, which means that there is greater volatility. So the difference between the geometric mean and the arithmetic mean returns would also be the varying amount. Now how to measure the historical means return for a portfolio of investment? In fact, mean returns of a portfolio are measured as the weighted average of the holding period yields for the individual investments in that particular portfolio. And for that purpose, we use the weights to determine averages and these weights are basically based on the market values of each investment. That is the reason these returns are termed as value weighted mean rate of return. In the example, we have three types of investment A, B and C. We have number of shares of these investments. We have share prices at the beginning of the investment period and we have ending prices of these investments for each share. Multiplying these individual share values with their investments each holding, we get the total amount of market value at the beginning and at the ending period. Using these values, we can determine the holding period return like we divide this 1.2 million over this 1 million. The holding period return is 1.2 and using this holding period return, we can determine the holding period yield. Now, how to determine the market weights? These will be based on these ending market values. If we divide this 1.2 million over 21.9 million, the resulting figure is 0.05. And if we divide this 16.5 million over this total investment portfolio of 21.9 million, the resulting figure is 0.75. To determine the weighted holding period yield, we multiply the individual investments holding period yield over its weight in the total basket. The total weighted holding period yield is now 0.095. So, to determine the portfolio's overall holding period return, we divide the portfolio's total ending market value over its total beginning value. The resulting holding period return is 1.09%. And when we deduct 1 from this 1.09% to determine the holding period yield on this portfolio, the resulting holding period yield is 9.5%. So, this is the rate of return that this particular portfolio has earned on this portfolio.